Finite Group Databasebeta

← #27272245

Subgroups — #27272245

Order 13568.

408 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274858C211yesmaximal, characteristic
2#20274464C211yesmaximal, characteristic
3#20272541C212yesmaximal
4#20272538C212yesmaximal
5C53×#2295C211yesmaximal, characteristic
6C2×#13133128—21no—
7#16810378—21no—
8#16810378—21no—
9#16810378—21no—
10#16810378C2212yes—
11#16810378C2212yes—
12C2×C4×Q8×C53C411yescharacteristic
13C53×#391C411yescharacteristic
14C53×#391C2211yescharacteristic
15C53×#389C411yescharacteristic
16C53×#389C2211yescharacteristic
17C53×#387—21no—
18C53×#387—21no—
19C4×C4⋊C4×C53—21no—
20C4×C4⋊C4×C53C411yescharacteristic
21C53×#376—21no—
22C43×C53C2211yescharacteristic
23C22×#88322—22no—
24C22×#88322—22no—
25#13133128—24no—
26C22×Q8×C53C2×C411yescharacteristic
27C4×Q8×C53—22no—
28C4×Q8×C53—22no—
29C2×C4⋊C4×C53—21no—
30C2×C4⋊C4×C53—21no—
31C2×C4⋊C4×C53—21no—
32C2×C4⋊C4×C53C2×C411yescharacteristic
33C2×C4⋊C4×C53C2×C411yescharacteristic
34C2×C4⋊C4×C53C2×C411yescharacteristic
35C2×C42×C53—21no—
36C2×C42×C53—21no—
37C2×C42×C53C2×C411yescharacteristic
38C2×C42×C53C2×C411yescharacteristic
39C2×C42×C53D411yescharacteristic
40C2×C42×C53D411yescharacteristic
41C2×C42×C53D411yescharacteristic
42C2×C42×C53D411yescharacteristic
43C2×C42×C53C2311yescharacteristic
44#95×C53—21no—
45#95×C53—21no—
46#95×C53—21no—
47#95×C53—21no—
48#95×C53—21no—
49#95×C53—21no—
50#95×C53—21no—
51C2×Q8×C53—21no—
52C2×Q8×C53—41no—
53C2×Q8×C53—41no—
54C2×Q8×C53M1612yes—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C4⋊C4×C53—24no—
67C4⋊C4×C53—21no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—41no—
70C4⋊C4×C53—41no—
71C4⋊C4×C53—42no—
72C4⋊C4×C53M1612yes—
73C42×C53—24no—
74C42×C53—22no—
75C42×C53—21no—
76C42×C53—21no—
77C42×C53—22no—
78C42×C53—22no—
79C42×C53—41no—
80C42×C53—41no—
81C42×C53—41no—
82C42×C53—42no—
83C2×#88322—24no—
84C2×#88322—42no—
85C2×#88322—42no—
86C2×#88322—42no—
87C2×#88322—42no—
88C2×#88322—42no—
89C23×C53C2×C22⋊C411yescharacteristic
90Q8×C53—42no—
91Q8×C53—42no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53#10412yes—
122C2×C4×C53C2×M1611yescharacteristic
123C2×C4×C53C2×M1611yescharacteristic
124C2×C4×C53#13512yes—
125C2×C4×C53#13712yes—
126#88322—44no—
127#88322—82no—
128#7611—531nomaximal, Sylow
129C22×C53#40711yescharacteristic
130C22×C53#41011yescharacteristic
131C22×C53#41311yescharacteristic
132C22×C53#41611yescharacteristic
133C22×C53#41911yescharacteristic
134C22×C53C2×#10411yescharacteristic
135C22×C53#42111yescharacteristic
136C4×C53—21no—
137C4×C53—21no—
138C4×C53—22no—
139C4×C53—22no—
140C4×C53—22no—
141C4×C53—41no—
142C4×C53—42no—
143C4×C53—41no—
144C4×C53—42no—
145C4×C53—41no—
146C4×C53—81no—
147C4×C53—81no—
148#2295D5311yescharacteristic
149#1839—531no—
150#1745—531no—
151#1272—532no—
152#1271—532no—
153C2×C53#146411yescharacteristic
154C2×C53#148811yescharacteristic
155C2×C53#149111yescharacteristic
156C2×C53#150011yescharacteristic
157C2×C53#151511yescharacteristic
158C2×C53#152911yescharacteristic
159C2×C53#155411yescharacteristic
160C2×C4×Q8Dic5311yescharacteristic
161C2×#97—1061no—
162#391Dic5311yescharacteristic
163#391C2×D5311yescharacteristic
164#389Dic5311yescharacteristic
165#389C2×D5311yescharacteristic
166#387—21no—
167#387—21no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4Dic5311yescharacteristic
170#376—21no—
171C43C2×D5311yescharacteristic
172#336—532no—
173#336—532no—
174#336—1061no—
175#336—1061no—
176#336—1061no—
177C53#761111yescharacteristic, complemented, Sylow
178C22×Q8C2×Dic5311yescharacteristic
179C22×C8—1062no—
180C22×C8—1062no—
181C4×Q8—22no—
182C4×Q8—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4C2×Dic5311yescharacteristic
187C2×C4⋊C4C2×Dic5311yescharacteristic
188C2×C4⋊C4C2×Dic5311yescharacteristic
189C2×C42—21no—
190C2×C42—21no—
191C2×C42C2×Dic5311yescharacteristic
192C2×C42C2×Dic5311yescharacteristic
193C2×C42C22⋊D5311yescharacteristic
194C2×C42C22⋊D5311yescharacteristic
195C2×C42C22⋊D5311yescharacteristic
196C2×C42C22⋊D5311yescharacteristic
197C2×C42C22×D5311yescharacteristic
198#97—1064no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95—21no—
203#95—21no—
204#95—21no—
205#95—21no—
206C2×Q8—21no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8#1171991712yes—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4C22⋊Dic5311yescharacteristic
215C22×C4C22⋊Dic5311yescharacteristic
216C22×C4C22⋊Dic5311yescharacteristic
217C22×C4C22⋊Dic5311yescharacteristic
218C22×C4C22×Dic5311yescharacteristic
219C22×C4C2×C22⋊D5311yescharacteristic
220C22×C4C2×C22⋊D5311yescharacteristic
221C2×C8—1064no—
222C2×C8—2122no—
223C2×C8—2122no—
224C2×C8—2122no—
225C2×C8—2122no—
226C2×C8—2122no—
227C4⋊C4—24no—
228C4⋊C4—21no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4#1171991712yes—
234C42—24no—
235C42—22no—
236C42—21no—
237C42—21no—
238C42—22no—
239C42—22no—
240C42—41no—
241C42—41no—
242C42—41no—
243C42—42no—
244C23C2×C22⋊Dic5311yescharacteristic
245Q8—42no—
246Q8—42no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4#1313314412yes—
277C2×C4C2×#1171991711yescharacteristic
278C2×C4C2×#1171991711yescharacteristic
279C2×C4#1313323712yes—
280C2×C4#1313324812yes—
281C8—4242no—
282C8—2124no—
283C22#1681058711yescharacteristic
284C22#1681059711yescharacteristic
285C22#1681060711yescharacteristic
286C22#1681061911yescharacteristic
287C22#1681063211yescharacteristic
288C22C2×#1313314411yescharacteristic
289C22#1681064011yescharacteristic
290C4—22no—
291C4—22no—
292C4—22no—
293C4—21no—
294C4—21no—
295C4—41no—
296C4—42no—
297C4—41no—
298C4—41no—
299C4—42no—
300C4—81no—
301C4—81no—
302C2#2027304911yescharacteristic
303C2#2027314911yescharacteristic
304C2#2027316211yescharacteristic
305C2#2027320111yescharacteristic
306C2#2027326611yescharacteristic
307C2#2027332211yescharacteristic
308C2#2027342511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2295C2
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2295D53
hypocenterC53#7611
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C43 ◁ #2295

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
177. C53#7611semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202748582
#202744642
#202725412
#202725382
C53×#22952
#761153